The 32-year-old England international had previously been linked with a move to Anfield, but both clubs currently remain silent on the move.
Lambert has scored 60 goals in the last three seasons at Southampton and is known to be a Liverpool supporter.
And Southampton legend and Sky Sports expert Matt Le Tissier has already suggested Lambert should be allowed to follow his heart if it tells him to play for his boyhood club.
He tweeted: "If Rickie joins lfc I will say good luck to him and thanks for helping to resurrect #sfc it's his boyhood team and in CL.#stillahero."
Sky Sports also brought you the news that Liverpool had made a bid for Lambert's Southampton and England team-mate Adam Lallana earlier this month.
The striker, who has scored 117 goals in 235 appearances for Southampton, is currently training with England ahead of the pre-World Cup friendly with Peru at Wembley on Friday.
Sky Bet had suspended betting on Lambert's future earlier on Thursday after he was heavily backed in from 5/1 for a move to Liverpool.
By the afternoon the striker had been backed in to 1/3 and went as short as 1/6 for a move to Anfield, at which point Sky Bet suspended the market with the betting trend failing to subside.
Sky Bet's head of football trading Paul Wiggins said: "We've seen a remarkable run of bets on the move having only opened the market this morning.
"As yet, we're unsure why it's proved so popular, but we had no choice but to shut up shop as the market became too one sided."
Lambert has been at Liverpool before, joining the club as a 10-year-old and spending five years there before heading to Blackpool as a trainee.
Earlier in the day, Liverpool boss Brendan Rodgers said he does not envisage signing any players on loan again this summer.
Since taking charge at Anfield, Rodgers has regularly dipped into the loan market to bolster his squad, with the likes of Nuri Sahin, Victor Moses and Aly Cissokho all spending time on Merseyside.
But now, with a UEFA Champions League campaign on the horizon next term, Rodgers does not believe loan signings will be beneficial to his squad.
"I would hope we don't have to go for loans this summer," Rodgers told the Liverpool Echo.
"But it's always about the availability of players. If you are not satisfied with the squad and you feel you can improve it, then you have to look at every option.
"In these first couple of years I've been here, we have had to do that as we've had a thin squad.
"The players we wanted we couldn't get but we're hopeful that this summer will be different. We turn the page and it's another chapter in terms of where we are going.
"Hopefully that will mean it will be more permanent players than loan players but I can never rule it out completely. There could be a situation where a player becomes available, we can get him and he can add to what we're doing."